This is question 8.16 from McCullagh and Nelder's book, Generalized Linear Models, 2nd Edition.
Suppose that $Y_1, \ldots, Y_n$ are independent and identically distributed with gamma distribution $G(\mu, \nu)$, both parameters taken to be unknown.
Here I'll note that they write the density function for $G(\mu, \nu)$ as
$$(\Gamma(\nu))^{-1} \left(\frac{\nu y}{\mu}\right)^\nu \text{exp}\left(-\frac{\nu y}{\mu}\right) d(\log y)$$
Note that here, $\mu$ is the mean of the random variable, and $\nu$ is a shape parameter.
Let $\bar{Y}$ be the arithmetic mean of the observations, and $\dot{Y}$ the geometric mean. Show that under the composite null hypothesis $\mathcal{H}_0: \mu = \mu_0$, $S_0 = \log (\dot{Y}/\mu_0) - \bar{Y}/\mu_0$ is a complete sufficient statistic for $\nu$.
This is not too hard, as one need only take the derivative of the log likelihood function and set it equal to zero. $\mu = \mu_0$ is treated as any other constant, so one finds that $S_0 = F(\nu)$ (using $F$ for some nontrivial expression involving $\nu$), which is what's needed.
What I need help with is what follows.
Show, again under $\mathcal{H}_0$, that the conditional joint distribution of $Z_i = \log(Y_i/\mu_0)$ given $S_0$ is uniform over the surface
$$\sum_i (z_i - \text{exp}(z_i)) = n S_0$$
Discuss briefly how you might use this result (i) to construct an exact test of $\mathcal{H}_0$, and (ii) to construct an exact confidence interval for $\mu$.
I have no idea how to show that the joint distribution given the sufficient statistic is uniform, as requested. And I'm not sure how exactly this can be used to construct a test for the null hypothesis. I could imagine how the test could be used for a confidence interval, though; given the data, the CI would consist of all $\mu_0$ for which the test would not reject the null hypothesis at the desired level of significance, which is one minus the confidence level.